Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13512 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2085 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | #135 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
482 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(6,13)(7,18,22,25)(8,11)(9,24)(10,23)(14,30,32,19)(15,34)(17,26)(20,36,27,21)(28,31,33,35);(5,6,8,13,12,17,11,26)(7,19,20,31,22,30,27,35)(9,16,10,15,23,29,24,34)(14,25,28,36,32,18,33,21);(2,4)(5,7,8,20,12,22,11,27)(6,14,13,28,17,32,26,33)(9,21,10,25,23,36,24,18)(15,19,29,31,34,30,16,35)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=d,d^2=i,e^2=h,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,f^a=fhi,h^a=hi,c^b=cf,e^b=ehi,f^b=fh,h^b=hi,e^c=eh,f^c=fh,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72239853, #72239855, #72240850, … |
| maximal normal subgroup of | #72239853, #72239855, #72240850, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55971);